A systematic small-rho expansion around zero-leverage stochastic-clock models yields fast semi-analytic formulas for barrier option prices and extrema distributions under leverage.
and Boenkost, W., The Heston stochastic volatility model with piecewise constant parameters - efficient calibration and pricing of window barrier options
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Extrema, Barrier Options, and Semi-Analytic Leverage Corrections in Stochastic-Clock Volatility Models
A systematic small-rho expansion around zero-leverage stochastic-clock models yields fast semi-analytic formulas for barrier option prices and extrema distributions under leverage.